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 A070079 a(n) gives the odd leg of the unique primitive Pythagorean triangle with hypotenuse A002144(n). 14
 3, 5, 15, 21, 35, 9, 45, 11, 55, 39, 65, 99, 91, 15, 105, 51, 85, 165, 19, 95, 195, 221, 105, 209, 255, 69, 115, 231, 285, 25, 75, 175, 299, 225, 275, 189, 325, 399, 391, 29, 145, 351, 425, 261, 459, 279, 341, 165, 231, 575, 465, 551, 35, 105, 609, 315, 589, 385, 675 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Consider sequence A002144 of primes congruent to 1 (mod 4) and equal to x^2 + y^2, with y>x given by A002330 and A002331; sequence gives values y^2 - x^2. Odd legs of primitive Pythagorean triangles with unique (prime) hypotenuse (A002144), sorted on the latter. Corresponding even legs are given by 4*A070151 (or A145046). - Lekraj Beedassy, Jul 22 2005 LINKS T. D. Noe, Table of n, a(n) for n=1..1000 FORMULA a(n)=A079886(n)*A079887(n) - Benoit Cloitre, Jan 13 2003 a(n) is the odd positive integer with A080109(n) = A002144(n)^2 = a(n)^2 + (4*A070151(n))^2, in this unique decomposition into positive squares (up to order). See the Lekraj Beedassy, comment. - Wolfdieter Lang, Jan 13 2015 EXAMPLE The following table shows the relationship between several closely related sequences: Here p = A002144 = primes == 1 mod 4, p = a^2+b^2 with a < b; a = A002331, b = A002330, t_1 = ab/2 = A070151; p^2 = c^2+d^2 with c < d; c = A002366, d = A002365, t_2 = 2ab = A145046, t_3 = b^2-a^2 = A070079, with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2). --------------------------------- .p..a..b..t_1..c...d.t_2.t_3..t_4 --------------------------------- .5..1..2...1...3...4...4...3....6 13..2..3...3...5..12..12...5...30 17..1..4...2...8..15...8..15...60 29..2..5...5..20..21..20..21..210 37..1..6...3..12..35..12..35..210 41..4..5..10...9..40..40...9..180 53..2..7...7..28..45..28..45..630 ................................. MATHEMATICA pp = Select[ Range[200] // Prime, Mod[#, 4] == 1 &]; f[p_] := y^2 - x^2 /. ToRules[ Reduce[0 <= x <= y && p == x^2 + y^2, {x, y}, Integers]]; A070079 = f /@ pp (* Jean-François Alcover, Jan 15 2015 *) CROSSREFS Cf. A002144, A002330, A002331, A080109, A070151 Sequence in context: A165260 A201874 A059528 * A142717 A057742 A201433 Adjacent sequences: A070076 A070077 A070078 * A070080 A070081 A070082 KEYWORD easy,nonn AUTHOR Lekraj Beedassy, May 06 2002 EXTENSIONS More terms from Benoit Cloitre, Jan 13 2003 Edited: Used a different name and moved old name to the comment section. - Wolfdieter Lang, Jan 13 2015 STATUS approved

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Last modified September 18 11:24 EDT 2024. Contains 376000 sequences. (Running on oeis4.)